Compactness and Rigidity of $\lambda$-Surfaces

TL;DR

This paper establishes a compactness theorem for $\lambda$-surfaces in $\mathbb{R}^3$ with bounded geometric properties, generalizing known results for self-shrinkers and leading to a rigidity theorem for convex cases.

Contribution

It introduces a generalized compactness theorem for $\lambda$-surfaces with uniform bounds, extending previous work on self-shrinkers and providing new rigidity results.

Findings

Proved a compactness theorem for $\lambda$-surfaces with uniform bounds.

Generalized Colding-Minicozzi's theorem for self-shrinkers.

Established a rigidity theorem for convex $\lambda$-surfaces.

Abstract

In this paper we develop the compactness theorem for $\lambda$-surface in $\mathbb R^3$ with uniform $\lambda$, genus, and area growth. This theorem can be viewed as a generalization of Colding-Minicozzi's compactness theorem for self-shrinkers in $\mathbb R^3$. As an application of this compactness theorem, we prove a rigidity theorem for convex $\lambda$-surfaces.